Add compound matrices: multiplicativity and entrywise nonnegativity - #408
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Step 3 (tractable half) of issue #394's decomposition; issue #405. Defines Matrix.compound q A, the matrix of all q x q minors indexed by increasing q-element selections, and proves compound_mul, that compound q (L * A) = compound q L * compound q A. That is exactly the existing Cauchy-Binet expansion det_submatrix_mul_eq_sum_powersetCard re-indexed, so the proof is three lines. Also proves compound_nonneg, that a totally nonnegative matrix has an entrywise nonnegative compound matrix, and compound_one_apply identifying the first compound matrix with A itself. Placed in the Mathlib mirror tree so it is upstream-shaped. Not proved here: that the eigenvalues of compound q A are the q-fold products of the eigenvalues of A, which is the hard half of #405 and the remaining spectral input for Gantmacher-Krein. No sorry and no axiom. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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Items 1 and 2 of #405 (does not close it — item 3, the eigenvalue-product
theorem, is the remaining hard half).
Adds
RealRooted/Mathlib/LinearAlgebra/Matrix/Compound.lean, mirroring theMathlib path for upstreaming:
compound q A: theq-th compound matrix, indexed bySet.powersetCard (Fin n) q, entries theq × qminors via the strictlymonotone enumeration
powersetEnum;compound_mul: multiplicativitycompound q (A * B) = compound q A * compound q B,a re-indexing of
Matrix.det_submatrix_mul_eq_sum_powersetCard(
Determinant/CauchyBinet.lean);compound_nonneg: total nonnegativity gives entrywise nonnegative compounds;compound_one_apply: theq = 1compound recovers the matrix.Sorry-free; builds clean on top of #403.
🤖 Generated with Claude Code